Question

Show that every polynomial of degree 1, 2, or 4 in ${\mathrm{\mathbb{Z}}}_2\left[x\right]$has a root in$\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^4+x+1\right)$}\right.$.

Short answer

It is proved every polynomial of degree 1, 2, or 4 in${\mathrm{\mathbb{Z}}}_2\left[x\right]$ has a root in$\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^4+x+1\right)$}\right.$ .

Step-by-step solution

Theorem 5.11

Theorem 5.11states that, for a field$F$ and irreducible polynomial$p\left(x\right)$ in $F\left[x\right]$,$\raisebox{1ex}{$F\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(p\left(x\right)\right)$}\right.$ will be an extension field $K$ of $F$, which contains a root of $f\left(x\right)$.

Show that every polynomial of degree 1, 2, or 4 in ℤ2x has a root in ℤ2xx4+x+1

Every irreducible polynomial over${\mathrm{\mathbb{Z}}}_2$ of degrees 1, 2, and 4 contains roots in $\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^4+x+1\right)$}\right.$, which is demonstrated as follows:

The polynomials$x$ and $x+1$ contain roots$\left[0\right]$ and $\left[1\right]$.

The polynomials$x^2+x+1^{}$ contain the root $\left[x^2+x+1\right]$.

The polynomials$x^4+x+1$ contain the root $\left[x\right]$.

The polynomials$x^4+x^2+1$ contain the root $\left[x^2+x+1\right]$.

The polynomials$x^4+x^3+1$ contain the root $\left[x^3+1\right]$.

The polynomials$x^4+x^3+x^2+x+1$ contain the root $\left[x^3\right]$.

All of these irreducible polynomials have degrees of 1,2, and 4.

Hence, it is proved every polynomial of degree 1,2, or 4 in${\mathrm{\mathbb{Z}}}_2\left[x\right]$ has a root in $\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^4+x+1\right)$}\right.$.